mass of the tip magnet. When the tip magnet passes by the coils some electromagnetic energy conversion occurs. The electromagnetic coupling can be characterized by the coupling coefficient, . When the tip magnet passes by the coils with the velocity , a force of magnitude impedes the motion of tip magnet. The current in the coils is . At the same time a potential difference is generated across the coil which equals . The charge passing through the coils is noted by and the overall inductance of the coils is . The following two terms in the Lagrangian represent the electromechanical energy in the coils: (6) The Euler-Lagrange equations for our three degrees of freedom system is: (7) The damping coefficient of the mechanical spring is denoted by . Performing the derivations in Eq. (7),dividing by the modal mass and grouping the terms results: (8) The coefficients in Eq. (8) are: , , , , , , , , (9) The second and third terms on the right hand side of Eq. (8-a) represent the “drag” terms introduced by the piezoelectric patch and the electromagnetic coils. The energy transferred to the electric circuits reduces the mechanical energy of the beam and therefore slightly suppresses its oscillations. The sign of the linear restoring coefficient, k, can be positive or negative. The familiar positive coefficient corresponds to low magnetic forces. In this situation 465
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